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A Parallel Splitting Method for Separable Convex Programs

Author

Listed:
  • K. Wang

    (Nanjing Normal University)

  • D. R. Han

    (Nanjing Normal University)

  • L. L. Xu

    (Nanjing Normal University)

Abstract

In this paper, we propose a new parallel splitting augmented Lagrangian method for solving the nonlinear programs where the objective function is separable with three operators and the constraint is linear. The method is an improvement of the method of He (Comput. Optim. Appl., 2(42):195–212, 2009), where we generate a predictor using the same parallel splitting augmented Lagrangian scheme as that in He (Comput. Optim. Appl., 2(42):195–212, 2009), while adopting a new strategy to get the next iterate. Under the mild assumptions of convexity of the underlying mappings and the non-emptiness of the solution set, we prove that the proposed algorithm is globally convergent. We apply the new method in the area of image processing and to solve some quadratic programming problems. The preliminary numerical results indicate that the new method is efficient.

Suggested Citation

  • K. Wang & D. R. Han & L. L. Xu, 2013. "A Parallel Splitting Method for Separable Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 138-158, October.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:1:d:10.1007_s10957-013-0277-9
    DOI: 10.1007/s10957-013-0277-9
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    References listed on IDEAS

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    1. Sun, Jie & Zhang, Su, 2010. "A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1210-1220, December.
    2. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    3. Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    4. Deren Han & Xiaoming Yuan & Wenxing Zhang & Xingju Cai, 2013. "An ADM-based splitting method for separable convex programming," Computational Optimization and Applications, Springer, vol. 54(2), pages 343-369, March.
    5. B. S. He & L. Z. Liao, 2002. "Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 111-128, January.
    6. Bing-Sheng He, 2009. "Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities," Computational Optimization and Applications, Springer, vol. 42(2), pages 195-212, March.
    7. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
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    Cited by:

    1. Yaning Jiang & Deren Han & Xingju Cai, 2022. "An efficient partial parallel method with scaling step size strategy for three-block convex optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 383-419, December.

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